Screw axes

Screws, the mathematical objects used to
describe rigid body motions and forces, are intrinsically linked with axes, be
it axes of motion (such as the rotation axis of a pure rotation) or force (such
as the line of action of a force).

Central and noncentral axes

The central axis of a screw \(\hat\bfs\) is the axis directed by the resultant
\(\bfr\) where the moment \(\bfm_P\) is parallel to \(\bfr\) (not necessarily
zero). A noncentral axis of the screw is any other axis in space directed by
the resultant \(\bfr\). From Varignon's formula, any two points \(P\) and \(Q\)
belonging to a noncentral axis have the same moment \(\bfm_P = \bfm_Q\). This
way, the moment field can be seen as a vector field over noncentral axes (5D
space) rather than over points (6D space).

Another interesting value for a screw is its automoment, which is the scalar
product \((\bfr \cdot \bfm_P)\) between its resultant and its moment. Again, from
Varignon's formula, it does not depend on the point \(P \in \mathsf{E}^3\)
selected to compute it. In particular, on the central axis \(\Delta_c\), the
moment is given by:

Illustration

Here is a small Python function with
mplot3d to plot the
moment field of a screw, assuming \(\Delta_c\) goes through the origin:

frommatplotlib.pyplotimportfigure,show,ionfrommpl_toolkits.mplot3dimportAxes3Dfromnumpyimportlinspace,cos,sin,arange,pi,array,crossdefplot_screw(res,pitch,scale=0.2):res=array(res)# just in casefig=figure()ax=fig.gca(projection='3d')forrinlinspace(0.,1.,4):forthetainlinspace(0,2*pi,20):x,y=r*cos(theta),r*sin(theta)ax.plot([x-res[0],x+res[0]],[y-res[1],y+res[1]],[-1,1],color='#999999')forzinarange(-1.,1.,0.3):u,v,w=scale*(cross([x,y,z],res)+pitch*res)ax.plot([x,x+u],[y,y+v],[z,z+w],color='b')ax.plot([-res[0],+res[0]],[-res[1],+res[1]],[-1,1],color='k',lw=5)show()

Here is the example of a screw with zero pitch, such as a pure rotation. The
bold black line shows the central axis, while the thin gray lines are for
noncentral ones. The moment field (linear velocities) is in blue.

Now, the same with a pitch \(\rho(\hat\bft) = 1\) m/rad, meaning each radian of rotation
brings a one meter translation:

We see where the word “screw” comes from: following the moment
field of the screw above is like following the thread of an actual
screw.